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Abstract—This article presents closed form expressions for the

self- and mutual inductances of circular wire wound coils used in

near field wireless power transfer systems. The calculation of the

radius of the coils, inspired from the Archimedean spiral found in

many biological organisms, is used to model the self-inductance of

single and multi-layer spiral coils. The value of the mutual

inductance is determined by expressing the Taylor expansion of

the Neumann’s integral for constant current carrying wires.

Formulas for mutual inductance are also derived for misaligned

magnetically coupled coils enabling the rapid but accurate

calculation of power transfer efficiency in real-life applications.

Self- and mutual inductance values are computed using the 3D

electromagnetic software package MAXWELL 3DTM and these

values demonstrate excellent agreement compared with the

proposed models. Wire wound coils of different geometrical

configurations have been manufactured to validate

experimentally the accuracy of the proposed models. Comparison

of analytical and experimental results indicate that the proposed

models are capable to accurately predict the self-inductance and

mutual coupling rapidly. The proposed modeling paves the way

for the time efficient optimization of near field wireless power

transfer links.

Index Terms—3D EM solver, multi-layer coil, misalignment,

near field wireless power transfer, single-layer coil.

I. INTRODUCTION

IRELESS power transfer is utilized today in a wide

range of applications, ranging from sophisticated

low-power biomedical implants [1]–[4] to high-power electric

vehicles [5]–[8]. The successful implementation of a wireless

power link depends to some extent on the accurate modeling of

the self- and mutual inductances that contribute to the

optimization of the coupling coefficient between the primary

transmission coil and the secondary receiver coil [9]. Achieving

optimum power transfer efficiency and large tolerance to

misalignment is today the subject of much research efforts

[10]–[15].

Several established methods for the calculation of the

self-inductance exist in the literature [10],[16]–[25]. Most of

these methods consider the current carrying loop as a set of

concentric

circles. Consequently, there is always some discrepancy

between the analytical expression of the self-inductance and the

simulated and experimental results. Many contributions in

literature have also been made to calculate the mutual

inductance [26]–[29], predominantly in the case of perfectly

aligned coils [30]–[32]. Analytical derivation of the mutual

inductance for misaligned coils is however rare [24],[33],[34]

and only carried out for translational misalignment despite the

fact that angular misalignment in some applications can also

drastically affect the mutual inductance and thus the power link

performance. A complete analysis of the variation of the mutual

inductance value that takes into account translational and

angular misalignment is therefore needed for the accurate

characterization of the performance of the wireless power

transfer link.

3D electromagnetic (EM) field solution software is one of

the most consistent approaches for computing self- and mutual

inductance values [33]. However, the simulation of complex

multi-loop and multi-planar wire wound coils (WWC) in a 3D

EM solver requires substantial computational time. On the

other hand, the previous semi-analytical approaches referenced

above require long numerical operations which do not reduce

the computational complexity [20],[26],[29],[30],[32]. In the

light of these difficulties, this article presents a simple, yet

accurate, method to estimate self- and mutual inductances to

design and optimize a near field wireless power transfer link. In

this study, the self-inductance is calculated using the

Archimedean spiral geometry used for the coiling of WWCs.

Furthermore, complex geometrical parameters are avoided and

computationally efficient models are presented for the

calculation of the mutual inductance. The proposed models

have been verified by 3D EM simulation and validated

experimentally. Various coil models have been designed and

simulated in ANSYS MAXWELL 3DTM

. The software package

is a commercial quasi-static solver [35], [36]. The

magnetostatic simulation mode has been used to solve the coil

models. The self-inductance of the same configurations has

been measured using a Fluke PM6306 RCL meter and an

auto-balancing bridge technique is adopted to measure the

mutual inductance of the physical coils. The experimental

Accurate Modeling of Coil Inductance for Near

Field Wireless Power Transfer

Sadeque Reza Khan, Sumanth Kumar Pavuluri and Marc P.Y. Desmulliez, Senior Member, IEEE

W

This study was financially supported by the UK Engineering & Physical

Research Council (EPSRC) under the programme grant Sonopill

(EP/K034537/2) and Heriot-Watt University through the Doctoral Training Account (DTA) programme for International Students.

The authors are with the Institute of Sensors, Signals and Systems, School of

Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, Scotland, UK (e-mail: [email protected]).


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values obtained indicate excellent agreement with the proposed

analytical models.

II. SELF-INDUCTANCE

The self-inductance, Lself, of a single circular coil of loop

radius, R , and wire diameter, w, can be represented as

[10],[20]–[25]

( ) [ (

) ] (1)

where µ is the permeability of the medium surrounding the coil.

For two perfectly aligned coils of radii Ri and Rj, with

center-to-centre distance, dij, the mutual inductance, Mij can be

calculated as [10], [20]

( )

√ *(

) ( ) ( )+

with

(2)

√

( )

(3)

where K(ij) and E(αij) are the complete elliptic integrals of the

first and second kind, respectively.

A. Self-inductance of a single layer spiral coil (SLSC)

The continuously varying radius of the loop of a spiral coil

needs to be calculated to determine the self-inductance of such

a coil. The coil is in fact an Archimedean spiral [37]. As shown

in Fig. 1, the calculated radius, R, is:

(4)

where R0 is the start radius, θR is the angle of revolution (=2j at

the jth

loop forming thereby the radius Rj) and sl is the space

between two consecutive loops.

Fig. 1. Parameters to calculate the varying radius of a 4-loop single layer

Archimedean spiral.

All the loops of a single-layer coil have a common center so

that dij=0. The self-inductance of Nl loops, Lself, can be

calculated from (1) and (2) as

∑ ( )

∑∑ ( ) ( )

(5)

where δij=1 for i=j; δij=0, otherwise.

The behaviour of the self-inductance is compared in Fig. 2

for different outer diameters (or number of loops) of a coil, for

wire diameter w=200 µm, starting radius R0=2.19 mm and

spacing sl=10 µm, against results obtained with the 3D-EM

field solution software package MAXWELL 3DTM

. The case of

the ideal coil of same outer diameter but with concentric circles

(labelled ideal CC) is also provided [10], [20]. The maximum

percentage of variation with respect to the 3D EM simulation is

also indicated on the same figure (right vertical axis). For the

proposed model, the variation is less than 3% whereas the use

of concentric circles produces a variation of around 45%.

Therefore, the proposed model offers better accuracy in the

approximation of Lself.

Fig. 2. Lself as a function of the outer diameter of the coil. The proposed new

model is benchmarked against the 3D EM modeling software package and the

idealized model where all circles are considered concentrical. The %variation with respect to the 3D EM values is also indicated for the ideal and proposed

models.

B. Self-inductance of a Multi-Layer Helical Coil (MLHC)

Fig. 3 shows the model of a multi-loop, multi-layer helical

coil where st represents the spacing between two consecutive

layers in the vertical direction. Li and Tj are the ith

loop and jth

layer, respectively.

Fig. 3. Multi-loop and multi-layer helical coil. The spacing between two loops

has been exaggerated to indicate the interconnection between the loops of the

MLHC.


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In this model, a loop L1 of radius R0 is formed initially with Tj

layers going downwards. At the bottom of the coil, a second

loop L2 of radius R1 is formed with Tj layers going upwards. At

the top of the coil, the same process continues for loop Li (= L3)

going downwards, etc. In this case, θR=2i–1) is used to

estimate the radius of the Lith

loop using equation (4). The

distance between the reference layer T1 and Tj, , is simply

( ) ( ) (6)

For a coil with Nt layers and Nl loops per layer, the total

self-inductance can be modeled as

where δij = 1 for i = j, δij = 0, otherwise. The first term in (7) is

the self-inductance of the Nt layers containing Nl loops each.

The second term accounts for the mutual inductance effect of

the loops for a single layer. The last term considers the mutual

inductance of different loops of different layers with one layer

always considered as a reference plane.

In Fig. 4, the performance of Lself, considered as an ideal CC

and MLHC is benchmarked against the 3D EM simulation as a

function of numbers of layers where Nl =2, w = 400 µm, D = 8.4

mm, st=10 µm and sl=10 µm. This configuration is

representative of biomedical implant applications such as

capsule endoscopy [38]. The variation with respect to the EM

software results, %variation, is around 12% for Nt=2,

decreasing to less than 5% for a larger number of layers. Again,

the proposed model shows better performance for a high

number of Nl and Nt.

Fig. 4. Lself for different layers Nt of multi-layer helical coil using 3D EM

simulation results, ideal (concentrical circles) model and proposed model.

III. MUTUAL INDUCTANCE OF INDUCTIVELY COUPLED COILS

In this section, the mutual inductance is modelled for

perfectly aligned and misaligned inductively coupled coils.

Results are compared with the 3D EM simulation.

A. Case of perfectly aligned coils

The mutual inductance, M, for two current carrying coils, C1

and C2, can be calculated using the Neumann’s formula [26],

[39]

∮∮

(8)

The magnitude, Rp, of the vector joining a point P on C1 to a

point Q lying on C2, as shown in Fig. 5, is:

[

( )]

(9)

where φC1 and φC2 are the angular co-ordinates of P on C1 and Q

on C2, respectively. RC1 and RC2 are the radii of the coils C1 of

wire diameter wC1, and C2 of wire diameter wC2, respectively.

The relative center-to-center distance, dr, between the coils

where these points lie, can be approximated as

( )

[( ) ( )]

[( ) ( )]

(10)

Fig. 5. Configuration of perfectly aligned coils. In this example, coil C1 has 5

loops and 4 layers. Coil C2 has 6 loops and 3 layers. Further, dr=d1 due to the

choice of the points P and Q.

where d1 is the center-to-center distance for the top and bottom

layers of C1 and C2, respectively. The point P lies on the Nt,C1th

layer in C1 taking the reference layer as the top layer in Fig. 5.

In the same way, the point Q lies in the Nt,C2th

layer in C2 taking

this time the bottom layer as the reference layer. Using (9) and

the dot product of the two infinitesimal displacement vectors,

dl1 and dl2, (8) can be rewritten as:

√

∫ ∫

( )

[ ( ]

(11)

∑ ( )

∑∑ ( ) ( )

∑ ∑ ∑ ( )

( )

( )

(7)


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where γ [33] is

(12)

Typical coils configurations used in WPT such as RC1=30

mm and RC2=4.2 mm and d1=100 mm, provide a value of <<1.

The integrand inside (11) can therefore be linearized as a

Taylor series up to the 5th

order such that

(

)

[

] (13)

For a coil C1 containing TC1 layers and LC1 loops and a coil

C2 containing TC2 layers and LC2 loops, the total mutual

inductance is then

Fig. 6. Mtotal at different d1 using 3D EM simulation results and proposed model. The percentage of variation with the software model is indicated on the right

vertical axis.

∑∑∑∑ ( ( ))

(14)

where RC1:k and RC2:l are the radius of kth

loop of C1 and lth

loop

of C2, respectively, and dr(i,j) is calculated according to (10).

Fig. 6 shows the value of the mutual inductance as a function

of d1 using the 3D EM simulation and (14). The coil radius for

the two coils was calculated using (4). The total numbers of

layers and loops were 12 and 1, respectively for C1, and 10 and

3, respectively for C2. The outer diameters for coils C1 and C2

were taken as 60 mm and 8.4 mm, respectively. The %variation

between 3D EM simulation and proposed model is less than 10

% for separation distance less than 70 mm and increases to

around 14% at d1=100 mm.

B. Case of translational misalignment

Fig. 7 shows the configuration between C1 and C2 in the

case of translational misalignment. C2 is off-axis by a distance,

d2, from the axis of the coil C1. The distance between two

arbitrary points P and Q lying on C1 and C2, respectively, can

be written as:

[

( ) ]

(15)

To calculate the mutual inductance, the following

parameters, γa, γb and γc are introduced:

(16)

(17)

(18)

Replacing (16), (17), (18) in (15) and using (8):

Fig. 7. Configuration of translational misalignment of C2. In this example, coil C1 has 5 loops and 4 layers. Coil C2 has 6 loops and 3 layers. Further, dr=d1 is

considered.

√

∫ ∫

( )

[ ( ) ]

(19)

The values of γa, γb and γc are less than unity for traditional

WWC configurations. Linearizing again the denominator of

(19) using a Taylor series expansion to the 4th

order and

integrating


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(

)

[

(

)

(

) (

)] (20)

where

(21)

Fig. 8. Mtotal at different translational displacements d2, for d1=50 mm using

3D EM simulation results and proposed model. The percentage of variation of the values of the proposed model with respect to the simulation results is

indicated on the right vertical axis.

Note that, when d2=0, which is the case of perfectly aligned

coils, the expression of M from (13) is recovered up to the third

order. The total mutual inductance, Mtotal, can be calculated as

∑∑∑∑ ( ( )

)

(22)

Fig. 8 shows the variation of Mtotal between C1 and C2 at

d1=50 mm for d2 ranging from 0 to 50 mm using the 3D EM

simulation software and (22). The coil parameters are the same

as in the case of perfectly aligned coils. The %variation

between 3D EM simulation and proposed model is less than

9%.

C. Case of angular misalignment

The configuration of coils suffering from angular

misalignment is shown in Fig. 9. Angular misalignment can be

due to roll and pitch rotations which are represented in the

figure by θ and λ, respectively. The distance between two

arbitrary point lying on C1 and C2 can be written as

*

( ) ( )

+

(23)

The parameters γa, γb and γc are introduced as

(24)

(25)

(26)

The values of γa, γb and γc are less than unity for wireless

power application. Replacing (24), (25) and (26) in (23) the

modified M from (8) can be written as

Fig. 9. Configuration of angular misalignment. In this example, coil C1 has 5

loops and 4 layers. Coil C2 has 6 loops and 3 layers. Further, dr=d1 is considered.

√

∫ ∫

( )

[ ( )

( )

]

(27)

The Taylor expansion of M can be written as:


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(

)

[

( )] (28)

Fig. 10. Mtotal at different θ for λ=45o and d1=50 mm using 3D EM simulation results and the proposed model. The percentage of variation of the values of

the proposed model with respect to the simulation results is indicated on the

right vertical axis.

Note that, when ==0, which is the case of perfectly

aligned coils, the expression of M from (13) is recovered up to

the third order. Mtotal can be written as

∑∑∑∑ ( ( )

)

(29)

Fig. 10 shows the changes of Mtotal for d1=50 mm as a

function of θ (with λ fixed at 45o) using 3D EM simulation and

(29) using the same coil parameters as in the previous section.

Allowing independent variations of θ and λ, the maximum %

variation with respect to the simulation model is also less than

7%. The changes of Mtotal as a function of θ and d1 (for λ=45o)

using (29) are provided in Fig.11. A 3D plot using the 3D EM

simulation provides similar results. The minimum and

maximum differences between the two surfaces occur at θ=60o

and d1=40 mm (4% variation) and θ=80o and d1=100 mm (13%

variation), respectively.

Fig. 11. Mtotal at different θ and d1 for λ=45o using the analytical model.

D. Case of translational and angular misalignment

The configuration in the case of combined translational and

angular misalignment is shown in Fig. 12. The distance

between two arbitrary points of C1 and C2 is approximated

such that:

Fig. 12. Configuration of the coil for translational and angular misalignment. In this example, coil C1 has 5 loops and 4 layers. Coil C2 has 6 loops and 3

layers. Further, dr=d1 is taken in to the consideration.

[

( )

( )

]

(30)

For simplification, the parameters γa, γb, γc, γd and γe are

introduced as

(31)

(32)

(33)

(34)

(35)

These parameters are smaller than unity for the wireless

power transfer coils. Replacing (31), (32), (33), (34) and (35) in

(30) the modified M from (8) can be written as


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√

∫ ∫

( )

[

( ) ( )

]

(36)

Finally, the Taylor expansion of M is

(

)

[

(

)

(

)

( )

( )

( ( ))

] (37)

where

(38)

(39)

Furthermore, Mtotal can be expressed as:

∑∑∑∑ ( ( )

)

(40)

Mtotal between C1 and C2 was calculated at d1=50 mm for d2

varying from 0 to 50 mm, and θ from 0 to 90o (with λ=45

o)

using (40) and 3D EM simulation. Paramters for the coils are

the same as in the previous section. The %variation between 3D

EM simulation and proposed model is less than 10% over the

complete range of d2 and angles.

IV. EXPERIMENTAL SETUP

As listed in Table I, various coils have been hand-made to

validate the proposed models using copper wire of different

wire diameters, number of layers and numbers of loops. A

Fluke PM6306 RCL meter was used to measure Lself of the

assembled coils.

Fig. 13. Schematic view of the experimental setup for the measurement of the

mutual inductance.

Fig. 13 shows the schematic of the experimental setup for

the measurement of the mutual inductance. Two sets of coils,

SLSC and MLHC, reported in Table I, are used as secondary

coils to experimentally validate the mutual inductance model.

The primary coil is a single layer coil made of 6 loops with

R=43.75 mm and w=1 mm. An auto-balancing bridge is used to

measure the mutual inductance. The voltage drop across the

primary coil is V1 and output voltage of the amplifier is VF. The

operational amplifier is connected in a differentiator

configuration and used as a negative impedance converter. The

amplifier sinks the input current of inverting port through RF

resistor connected in the output. I1 is the current through the

primary coil and induces a voltage V2 in the secondary side. The

transfer impedance seen at the secondary coil is

| | | | |

| (41)

Therefore,

| |

(42)

where f is the frequency of the input signal.

V. EXPERIMENTAL RESULTS

Fig. 14 shows the measured values of Lself for the SLSC

presented in Table I with varying outer diameters.

Fig. 14. Measured Lself values at different outer diameters for the single-layer WWC. The percentage of variation of the proposed model and

3D EM simulation with respect to the measured results is indicated on the

TABLE I

COIL PROPERTIES

Coil Type R (mm) w (μm) Nl Nt

Single loop single layer 30 1000 5 1

Single loop multi-layer 5.2 560 1 12

Multi-loop multi-layer 7.5 720 3 5


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right vertical axis.

The measured result is compared with the simulation and the

proposed model. The %variation of the model and the 3D EM

simulation with respect to measured results is presented on the

right vertical axis and show the %variation is less than 7% for

both the proposed model and 3D EM simulation

Fig. 15. Measured Lself values for multi-layer, single loop coils and

comparison with the proposed model and 3D EM simulation results. The percentage of variation of the values of the proposed model and 3D EM

simulation with respect to the measured results is indicated on the right

vertical axis.

Fig. 15 shows the measured values of Lself for the multi-layer

single loop case presented in Table I as the number of layers

vary. The %variation of the experimental results with respect to

both the proposed and 3D EM simulation models is less than

6%.

Fig. 16 presents the experimental and simulation results of

Lself for the multi-loop, multi-layer (Nt=5) coil as the number of

loops is varied from 1 to 5. The variation for both the proposed

and 3D EM simulation models with respect to experimental

values is less than 8%.

Fig. 16. Measured Lself values as a function of the number of loops for the

multi-loop (Nt =5) WWC and comparison with the proposed model and 3D

EM simulation results. The percentage of variation for the proposed model

and 3D EM simulation with respect to the measured results is indicated on

the right vertical axis.

Fig. 17 shows the measurement results in comparison with

(14) and 3D EM simulation for perfectly aligned single layer

primary and secondary coils. The %variation occurring with

respect to the proposed model is less than 3.5% for single layer

coils. Similar plot of multi-layer coils demonstrates less than

10% variation for the proposed model.

Fig. 17. Measurement of Mtotal at different d1 and comparison with proposed

model and 3D EM simulation results for perfectly aligned single layer primary and secondary coils. The %variation of the proposed model and 3D EM

simulation with respect to measured results are presented in the right vertical

axis.

Fig. 18 shows the magnetic field distribution of perfectly

aligned single layer primary and secondary coils. In Fig. 18 (a),

d1=10 mm, the secondary coil is strongly exposed to the

magnetic field radiation of the primary coil due to smaller

center-to-center distance. Figs. 18 (b) and (c) demonstrate

lower magnetic field intensity in the secondary coil due to the

larger separation distance (d1=35 and 50 mm, respectively) of

the primary coil. This confirms the reduction of Mtotal due to the

increase in d1.

(a)


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(b)

(c)

Fig. 18. Magnetic field distribution for perfectly aligned coils. (a) d1=10 mm. (b) d1=35 mm. (c) d1=50 mm.

Fig. 19 presents the measured and 3D EM simulated results

for different translational misalignment distances, d2, for

d1=10mm for single layer primary and secondary coils. The

variation in the proposed model remains less than 3%. The

same type of measurement for multi-layer coils can show a

%variation of slightly over 6%.

Figs. 20 (a) and (b) show the magnetic field distribution for

d2=10 and 30 mm, respectively, for d1=10 mm. The field

intensity in the secondary coil due to primary coil is lower in

both cases compared to Fig. 18 (a), causing thereby a reduction

of the Mtotal.

Fig. 19. Measurement of Mtotal values at different translational displacement d2 for d1 = 10 mm and comparison with the proposed model and 3D EM simulation

results for single layer primary and secondary coils. The %variation of the

proposed model and 3D EM simulation with respect to measured results are

presented in the right vertical axis.

(a)

(b)

Fig. 20. Magnetic field distribution for translational misalignment for d1 = 10

mm. (a) d2=10 mm. (b) d2=30 mm.

Fig. 21 shows the measured, calculated (29) and 3D EM

simulated results of angular misalignment for single-layer

primary and secondary coils. The roll rotation angle, θ is

changed for a constant pitch rotation angle λ=30o. The

%variation remains less than 6.5% for the proposed model and

3D EM simulation for single layer coils. In the case of

multi-layer coils, the percentage variation is around 8%.

Fig. 21. Measurement of Mtotal for single layer primary and secondary coils at

different θ for λ=30o and d1=35 mm. The proposed model and 3D EM simulation results are also compared. The %variation of the proposed model

with respect to measured results are presented in the right vertical axis.


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Further, Fig. 22 shows the measured, calculated (29) and 3D

EM simulated results of varying θ with respect to constant

λ=45o and d1=35 mm. The observed %variation is less than 7%

for the proposed model and 3D EM simulation for single layer

coils. In the case of similar plot of multi-layer coil, this

percentage variation is raised up to 10%.


different θ for λ=45o and d1=35 mm. The proposed model and 3D EM

simulation results are also compared. The %variation of the proposed model with respect to measured results are presented in the right vertical axis.

In Fig. 23, the measured, calculated (29) and 3D EM

simulated results are illustrated for varying θ with respect to

constant λ=60o

and d1=35 mm. The %variation is close to 5%

for the proposed model and 3D EM simulation for single layer

coils. This percentage variation is less than 11% for the similar

plot of multi-layer coil.


different θ for λ=60o and d1=35 mm. The proposed model and 3D EM

simulation results are also compared. The %variation of the proposed model with respect to measured results are presented in the right vertical axis.

Figs. 24 (a), (b), (c) and (d) show the magnetic field

distribution for θ=λ=30o, θ=λ=45

o, θ=λ=60

o and θ=λ=90

o,

respectively, for d1=35 mm. Compared to Fig. 18 (b) the

exposure of the magnetic radiation from primary coil reduces

with the gradually increasing θ and λ of the secondary coil.

Mtotal reduces therefore significantly. In case of θ=λ=90o, the

field vectors of primary coil are in parallel with the position of

the secondary coil bringing Mtotal close to zero.

(a)

(b)

(c)

TABLE II

COMPARISON OF THE SIMULATION TIMES

Configuration of the coil(s) Simulation Time

(3D EM)

Simulation Time

(Proposed Model)

Lself, single layer 10 min 12 × 10˗3 s Lself, single loop multi-layer 15 min 15 × 10˗3 s

Lself, multi-loop multi-layer 45 min 18 × 10˗3 s

Mtotal, perfectly aligned (single

layer secondary coil)

6 hr 30 min 2.3 s

Mtotal, translational misaligned (single layer secondary coil)

5 hr 45 min 2.1 s

M, angularly misaligned

(single layer secondary coil)

5hr 30 min 2.4 s

M, incorporated misaligned

(single layer secondary coil)

12 hr 40 min 5 s


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(d)

Fig. 24. Magnetic field distribution for angular misalignment at d1 = 35 mm. (a)

θ=λ=30o. (b) θ=λ=45o.(c) θ=λ=60o. (d) θ=λ=90o.

The calculation of Mtotal at different d2 and θ ranging from 0 to

90o (λ=45

o, d1=35 mm) for the single layer secondary coil has

also been conducted. The %variation of the measured results

with the proposed model (40) and 3D EM simulation are less

than 3% and 5.5%, respectively.

The proposed model is designed for near field wireless power

transfer applications. It is observed that the simulated M

depends on the frequency. This is mainly caused by the

parasitic effects near the self-resonant frequency of the coil

[33]. In Table I, the single layer coil has a self-resonance

frequency over 40 MHz. A variation of less than 10% between

the analytical and simulated M is recorded in the complete kHz

frequency range. This variation increases up to 15% at 20

MHz, 20% near 30 MHz and 34.5% at 40 MHz. The multi-layer

coil shows a variation over 25% near their self-resonant

frequency (over 30 MHz). It is therefore recommended that the

analytical model be used only up to half the self-resonant

frequency of the coil studied.

VI. COMPUTATIONAL EFFICIENCY

Table II provides a comparison of the computational time

required for the proposed analytical model (MATLABTM

) and

the 3D EM software (MAXWELL 3DTM

) using parameters for

the coils manufactured for this study. The simulation was

performed on an Intel (R) Xeon (R) CPU E5-2640 with

processor speed of 2.5 GHz and 128 GB of RAM (HP Z820

workstation). The comparison confirms that a significant

amount of simulation time can be saved using the models

proposed in this paper.

VII. CONCLUSIONS

This paper presents an accurate modeling method of the

self-inductance, and compact solutions for the calculation of

the mutual inductance for near field wireless power transfer

systems. The detailed mathematical derivation of inductances

is presented. The proposed models have been compared with

the 3D EM simulation results for numerical validation and

experimentally validated. This method shows excellent

prediction capability of self-inductance for all types of WWC

coils (single and multi-layer). Additionally, it provides a

cost-effective calculation for the determination of the mutual

inductance values at any separation distance and misalignment

cases. Self- and mutual inductance of different assembled coils

are measured and compared with the proposed prediction

models. Results demonstrate an excellent agreement in the

accuracy of the proposed models. The comparison of the

required computational time also proves the efficiency of the

proposed models. Therefore, the self- and mutual inductance

models presented in this paper can be considered as an

excellent candidate for the design and optimization of

sophisticated near field wireless power transfer systems.

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Sadeque Reza Khan received his BSc

degree in electronics and

telecommunication engineering from

University of Liberal Arts Bangladesh

on 2010 and M.Tech degree in VLSI

design from National Institute of

Technology Karnataka, India in 2014.

He is currently pursuing Ph.D. degree in

electrical engineering at Heriot-Watt

University, UK.

Sumanth Kumar Pavuluri received a

Ph.D. degree from Heriot-Watt

University, Edinburgh, UK, in 2011. He

is currently a post-doctoral research

associate at Heriot-Watt University

working on microwave sensing, micro

machined antennas, microwave

applicators for curing and heating

applications.

Marc Desmulliez (SM’87) received a

Ph.D. in optoelectronics form

Heriot-Watt University in 1995. He is

currently Professor of Microsystems

Engineering at Heriot-Watt University

and leads the Multi-modal Sensing and

Micro-Manipulation Research Group.

accurate modeling of coil inductance for near field ... · self-inductance can be modeled as where...

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